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The Energy Level of the Hydrogen EM-free Rotator Atom



The Energy Level of the Hydrogen EM-free Rotator Atom

When you are inter­ested in physics you must read “Unbe­liev­able”!

Although it was so far not pos­si­ble to obtain a math­e­mat­i­cal solu­tion for a EM-​free rota­tor that resem­bles the hydro­gen atom with it’s clear energy lev­els we con­tinue the search.

When the elec­tron cir­cles around the pro­ton at a smaller dis­tance (R2<R1) the elec­tro­sta­tic energy (We) of the sys­tem is decreased accord­ing to:

(2)

The kinetic energy of the hydro­gen atom, when con­sid­er­ing the atom is an EM-​rotator, would increase because the elec­tron is now cir­cling around with higher speed.

(3)

Con­sid­er­ing equa­tion (1):

we can express the dynamic/​kinetic energy of the sys­tem Wk (3) with:

(4)

Equa­tion (2) shows the dif­fer­ence in elec­tro­sta­tic energy lev­els between the orbit radius R2 and R1. The elec­tro­sta­tic energy level of the atom, when R1= and R2=Re, is:

(2a)

We observe that the kinetic energy of the sys­tem Wk (4) is at all times half of the released poten­tial energy We of the elec­tro­sta­tic field (2a). The total energy of the sys­tem is the poten­tial energy plus the kinetic energy:

(equa­tion 2a+ equa­tion 4)

(5)

When the atom emits a pho­ton in our EM-​rotator, due to the descent of the elec­tron to a lower orbit, the energy of the pho­ton is half the decreased poten­tial energy. Because of the energy con­ser­va­tion law the energy of the emit­ted pho­ton must be:

These are the pho­tons emit­ted by the EM-​rotator when the orbit­ing dis­tances Re deter­mines the energy level.

The energy level of the hydro­gen atom, accord­ing to the Bohr-​atomic model (Wb), is:

(6) where n is the prin­ci­pal quan­tum number.

The Bohr-​atomic model describes the observed energy lev­els of the atom very well for n=1,2,3… In equa­tion (6) the only vari­able is n. The energy level of the EM-​rotator (5), is com­pletely deter­mined by the dis­tance of the elec­tron to the nucleus.

In clas­si­cal EM-​physics the energy level of the atom is com­pletely deter­mined by the dis­tance between nucleus and elec­tron (5). The QM-​solution (6) shows a quan­ti­fied for­mula where the orbit­ing dis­tance is no longer pre­sented. Bohr’s Cor­re­spon­dence Prin­ci­ple tells us that both worlds (QM and EM) have to obey the same physic laws. How­ever the quan­tum rules are of no sig­nif­i­cance in the macro-​world.

At the quan­tum level physics have to obey the quantum-​physics laws and the macro­physics laws. The Cor­re­spon­dence Prin­ci­ple of Bohr tells us that at the quan­tum level there are no addi­tional rules, only that in the macro-​world the quan­tum rules are no longer sig­nif­i­cant. The same prin­ci­ple tells us that the macro­physics laws also have to be valid at the quan­tum level.

Although the equa­tions (5) en (6) are dif­fer­ent Bohr’s Cor­re­spon­dence Prin­ci­ple tells us they could be the same; (5) describ­ing the rules of the macro-​world and (6) the rules of the micro-​world where the laws of both worlds are relevant.

Neglect­ing the Me/​Mp fac­tor in equa­tion (5) we get:

(5a) (**)

where Rn is the orbit­ing dis­tance of the electron.

When equa­tions (5a) and (6) are equally valid we can express the orbit­ing dis­tance Re with the prin­ci­pal quan­tum number.

The energy of equa­tion (6) must be equal to equa­tion (5a) and therefore:

(5c)

(7)

For the ground level of the Bohr-​hydrogen atom (n=1) the cal­cu­lated dis­tance, of course, coin­cides with the Bohr radius of the hydro­gen atom Rb=5,29177.10^-11 meter; the orbit­ing dis­tance of the elec­tron being in ground state.

Sim­i­lar cal­cu­la­tions with the ryd­bergcon­stant and equa­tion (5c) gives:

(8)

The ryd­berg prin­ci­pal quan­tum num­ber is cal­cu­lated at .

The above cal­cu­la­tions of the Bohr-​radius of the hydro­gen atom (n=1) and the ryd­berg quan­tum num­ber are com­pletely con­sis­tent with QM-​calculations. Assum­ing that equa­tion (5a) is iden­ti­cal to (6) doesn’t imply any discrepancy.

Sum­ma­riz­ing we deducted that the EM-​free rota­tor for the hydro­gen atom has infi­nite solu­tions. With the assump­tion that the hydro­gen atom is an EM-​rotator there is at any dis­tance or speed a pos­si­ble equi­lib­rium. The dis­crete energy lev­els of the elec­trons in the atom, the quan­ti­sa­tion of energy, can com­pletely be explained by a quan­tum restric­tion that the elec­tron can only have sta­ble orbits around the nucleus at the dis­crete dis­tances; .

Next chap­ter: The Quan­ti­sa­tion of Distance

When you are interested in physics you must read “Unbelievable“!

Although it was so far not possible to obtain a mathematical solution for a EM-free rotator that resembles the hydrogen atom with it’s clear energy levels we continue the search.

When the electron circles around the proton at a smaller distance (R2<R1) the electrostatic energy (We) of the system is decreased according to:

  (2)

The kinetic energy of the hydrogen atom, when considering the atom is an EM-rotator, would increase because the electron is now circling around with higher speed.

 (3)

Considering equation (1):

we can express the dynamic/kinetic energy of the system Wk (3) with:

(4)

Equation (2) shows the difference in electrostatic energy levels between the orbit radius R2 and R1. The electrostatic energy level of the atom, when R1= and R2=Re, is:

(2a)

We observe that the kinetic energy of the system Wk (4) is at all times half of the released potential energy We of the electrostatic field (2a). The total energy of the system is the potential energy plus the kinetic energy:

 (equation 2a+ equation 4)

(5)

When the atom emits a photon in our EM-rotator, due to the descent of the electron to a lower orbit, the energy of the photon is half the decreased potential energy. Because of the energy conservation law the energy of the emitted photon must be:

These are the photons emitted by the EM-rotator when the orbiting distances Re determines the energy level.

The energy level of the hydrogen atom, according to the Bohr-atomic model (Wb), is:

 (6) where n is the principal quantum number.

The Bohr-atomic model describes the observed energy levels of the atom very well for n=1,2,3… In equation (6) the only variable is n. The energy level of the EM-rotator (5), is completely determined by the distance of the electron to the nucleus.

In classical EM-physics the energy level of the atom is completely determined by the distance between nucleus and electron (5). The QM-solution (6) shows a quantified formula where the orbiting distance is no longer presented. Bohr’s Correspondence Principle tells us that both worlds (QM and EM) have to obey the same physic laws. However the quantum rules are of no significance in the macro-world.

At the quantum level physics have to obey the quantum-physics laws and the macrophysics laws. The Correspondence Principle of Bohr tells us that at the quantum level there are no additional rules, only that in the macro-world the quantum rules are no longer significant. The same principle tells us that the macrophysics laws also have to be valid at the quantum level.

Although the equations (5) en (6) are different Bohr’s Correspondence Principle tells us they could be the same; (5) describing the rules of the macro-world and (6) the rules of the micro-world where the laws of both worlds are relevant.

Neglecting the Me/Mp factor in equation (5) we get:

      (5a)  (**)

where Rn is the orbiting distance of the electron.

When equations (5a) and (6) are equally valid we can express the orbiting distance Re with the principal quantum number.

The energy of equation (6) must be equal to equation (5a) and therefore:

 (5c)

 (7)

For the ground level of the Bohr-hydrogen atom (n=1) the calculated distance, of course, coincides with the Bohr radius of the hydrogen atom Rb=5,29177.10^-11  meter; the orbiting distance of the electron being in ground state.

Similar calculations with the rydbergconstant  and equation (5c) gives:

 (8)

The rydberg principal quantum number is calculated at .

The above calculations of the Bohr-radius of the hydrogen atom (n=1) and the rydberg quantum number are completely consistent with QM-calculations. Assuming that equation (5a) is identical to (6) doesn’t imply any discrepancy.

Summarizing we deducted that the EM-free rotator for the hydrogen atom has infinite solutions. With the assumption that the hydrogen atom is an EM-rotator there is at any distance or speed a possible equilibrium. The discrete energy levels of the electrons in the atom, the quantisation of energy, can completely be explained by a quantum restriction that the electron can only have stable orbits around the nucleus at the discrete distances; .

Next chapter: The Quantisation of Distance

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